Answer:
The number satisfies the conclusion of Rolle's Theorem for .
Step-by-step explanation:
According to the Rolle's Theorem, for all function continuous on , there is a value () such that:
(Eq. 1)
Where:
- First derivative of the function evaluated at , dimensionless.
, - Lower and upper bounds, dimensionless.
, - Function evaluated at lower and upper bounds, dimensionless.
Let , then upper and lower values are, respectively:
Lower bound ()
Upper bound ()
From Rolle's Theorem, we find that first derivative evaluated at is:
Then, we find the first derivative of the function, equalize to and solve the resulting expression:
The number satisfies the conclusion of Rolle's Theorem for .